Questions tagged [harmonic-analysis]

Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.

Filter by
Sorted by
Tagged with
2 votes
0 answers
117 views

Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?

Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary: $$ |f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg). $$ Does it follow that the (...
André Henriques's user avatar
3 votes
0 answers
73 views

Simple-looking inequality for Fourier series

I am looking for a reference (and possibly some history) for the following inequality for finite sums of complex exponentials. Let $+\infty>b>a>-\infty$. There exists some constant $C>0$ ...
char's user avatar
  • 309
3 votes
1 answer
147 views

How to compute the left-inverse of $f$ in the sense of $\approx$? [duplicate]

For two functions $\varphi,\psi:\Omega\times[0,\infty)\to[0,\infty)$, $\varphi\approx\psi$ means that there exist constants $c_1,c_2>0$, such that $\forall t\geq0$, $\forall x\in\Omega$, $$c_1\...
Wa haha's user avatar
  • 51
11 votes
1 answer
379 views

Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the ...
André Henriques's user avatar
2 votes
2 answers
239 views

Given a specific function $f$, how to compute the left-inverse of $f$ in the sense of $\approx$?

For a non-negative function $\varphi$ defined on $[0,\infty)$, the left-inverse $\varphi^{-1}$ of $\varphi$ is defined by setting, $\forall t\geq 0$, $$\varphi^{-1}(t):=\inf\{u\geq0:\varphi(u)\geq t\}....
Wa haha's user avatar
  • 51
7 votes
2 answers
931 views

$L^p$ bounds on tails of bounded $L^q$ sequences

Note: This is a generalisation of an earlier problem as suggested by user Jochen Glueck in the comments. Let $1 \leq p < q \leq \infty$, and $f_n: [0, 1] \to \mathbb R$ be a sequence of functions ...
Nate River's user avatar
  • 4,822
1 vote
2 answers
138 views

Extending a discrete singular kernel

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
A beginner mathmatician's user avatar
3 votes
1 answer
163 views

Discrete singular integrals

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
A beginner mathmatician's user avatar
-1 votes
1 answer
76 views

Fundamental of a signal

Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
Arthur B's user avatar
  • 1,882
3 votes
0 answers
118 views

Riesz potential on the boundary of a smooth domain

Let $\Omega \subseteq \mathbb{R}^n$ be a measurable set of finite measure. It is well-known that there holds $$ \sup_{x \in \mathbb{R}^n} \int_{\Omega} \frac{d z}{| x - z |^{n - 1}} \leqslant c_n | ...
Kosh's user avatar
  • 356
1 vote
1 answer
381 views

On level sets of smooth functions in a bounded domain

Let $\Omega$ be a bounded domain in $\mathbb R^n$, $n\geq 2$, with a smooth boundary and let $f$ be a smooth function on $\bar\Omega$. Is there a natural condition that one can impose on $f$ ( say in ...
alz812's user avatar
  • 11
8 votes
2 answers
759 views

Points where harmonic functions fail to give a coordinates system

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
Ali's user avatar
  • 4,077
9 votes
0 answers
329 views

Can one prove Rademacher’s theorem via the rising sun lemma?

The classical Rademacher’s theorem states that Lipschitz continuous functions on $\mathbb R^n$ are differentiable almost everywhere. In dimension one, a stronger result holds - it can be shown that ...
Nate River's user avatar
  • 4,822
20 votes
0 answers
301 views

Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
Calculix's user avatar
  • 321
4 votes
1 answer
373 views

Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation \...
asv's user avatar
  • 21.1k
1 vote
0 answers
135 views

Fourier transform of the Bochner-Riesz multipliers

How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define: $$ \hat{m_{\lambda}}(x)=\int\limits_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \...
Dapao Zhang's user avatar
3 votes
0 answers
91 views

About the nilpotency of a subgroup

Let $G$ be a compact group. Let $\mathcal N$ be a family of closed normal subgroups of nilpotency class at most $k$. Assume that $\mathcal N$ is closed under finite intersections and $\bigcap_{N\in\...
Meisam Soleimani Malekan's user avatar
1 vote
0 answers
212 views

On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
Dionel Jaime's user avatar
4 votes
0 answers
103 views

Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials? In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
Pierre's user avatar
  • 41
4 votes
1 answer
139 views

Can the degree of $k$-nilpotence of a simple simply connected compact Lie group be in $(0,1)$?

Let $G$ be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of $k$-nilpotence of $G$ to be the Haar measure of the set $$\{(x_1,\dotsc,x_{...
Meisam Soleimani Malekan's user avatar
-1 votes
1 answer
409 views

Harmonic function in infinite domain in $\mathbb{R}^3$, constant on the boundary and decaying as $1/r^2$

EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\...
asv's user avatar
  • 21.1k
0 votes
1 answer
378 views

Harmonic functions in infinite domain in Euclidean space

EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
asv's user avatar
  • 21.1k
3 votes
0 answers
197 views

The Fourier transform of a compactly supported smooth function on Lie groups over $\mathbb{Q}_S$, where $S$ contains finitely many primes and $\infty$

Let $G$ be a semisimple Lie group defined over global Field $\mathbb{Q}$. Let $S$ be a set of finitely many non-Archimedean places including Archimedean places. Let $P_{0}=M_{0}A_{0}N_{0}$ be the ...
user938363's user avatar
6 votes
1 answer
264 views

A property of rapid sequences of natural numbers

$\newcommand{\IR}{\mathbb R}$ $\newcommand{\IT}{\mathbb T}$ $\newcommand{\w}{\omega}$ $\newcommand{\e}{\varepsilon}$ Taras Banakh and me proceed a long quest answering a question of ougao at ...
Alex Ravsky's user avatar
  • 4,102
6 votes
1 answer
376 views

Absolute values of two functions and absolute values of their Fourier transform coincides

Let $f, g \in L^2(\mathbb{R})$. Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$? I am not able to prove it or ...
J.Mayol's user avatar
  • 489
2 votes
0 answers
109 views

Anticommutation of convolution products on trace class operators of quantum groups

This question was originally posted to MathStackExchange. Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
Ben A-S's user avatar
  • 59
4 votes
0 answers
119 views

Algebra properties regarding Gevrey spaces: closed under multiplication

In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
Feng's user avatar
  • 517
3 votes
1 answer
153 views

Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
BigbearZzz's user avatar
  • 1,245
15 votes
1 answer
460 views

For what LCH groups is the Haar measure $\mu(U x U)$ bounded?

Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function $$ \Phi: \quad G \to (0,\infty), \quad x \...
PhoemueX's user avatar
  • 754
0 votes
0 answers
79 views

An amenable operator algebra has the total reduction property

This is from https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
Korn's user avatar
  • 101
0 votes
0 answers
96 views

Integral kernel of resolvent of Sub-Laplacian?

Consider the Laplacian $-\Delta: C^\infty (S^1) \to C^\infty(S^1)$ where $\mathbb R/\mathbb Z=S^1$ and for a periodic function $f:\mathbb R\to\mathbb R$ we have $-\Delta f=-f''$. For the orthonormal ...
scroo0ooge's user avatar
8 votes
2 answers
592 views

Vanishing rate of a harmonic function near a boundary point

Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is, $$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$ for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
Jacob Lu's user avatar
  • 903
2 votes
0 answers
144 views

(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis

It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
BigbearZzz's user avatar
  • 1,245
12 votes
1 answer
693 views

A generalization of Rubio de Francia's inequality

Suppose that $\{I_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
Anton Tselishchev's user avatar
2 votes
1 answer
90 views

Joint boundedness of families of random Fourier series

Let $\varepsilon_n$, $ n \in \mathbb {Z}$, be independent Rademacher random variables. Is there a characterization of those sequences $a_{m,n}$, $(m,n) \in \mathbb {Z} \times \mathbb{Z}$, of complex ...
Rob Podolski's user avatar
2 votes
0 answers
57 views

Uniqueness for a certain semilinear equation

Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation $$ \begin{aligned} \begin{...
Ali's user avatar
  • 4,077
2 votes
0 answers
58 views

How to prove this this integral equality which contain nonlocal operator, $(-\partial_{xx})^{1/2}$?

Suppose that $\theta(t,x)$ is even about $x$ and is smooth. $0\le \gamma<1/2$, $0<\delta<1-2\gamma$. $\Lambda=(-\partial_{xx})^{1/2}$ My Question: How to prove that $$ \int_0^{\infty} \frac{(\...
Mr.xue's user avatar
  • 171
0 votes
1 answer
191 views

The Quotient exponential operator

I have a question if you don't mind. I have the following quotient operator: $$\frac{1}{e^{d/dx}(f(x))}$$ Where $f$ is a smooth function on $R$. I would like to get rid of the denominator. IS there ...
Adam Hammam's user avatar
1 vote
1 answer
312 views

How is the Cauchy-Schwarz inequality used to bound this derivative?

In "Hardy's Uncertainty Principle, Convexity and Schrödinger Evolutions" (link) on page 5, the authors state that they are using the Cauchy-Schwarz inequality to bound the derivative of the $...
Dispersion's user avatar
1 vote
0 answers
487 views

Can we generalize the concept of "characters" in group theory via methods from statistics and probability theory?

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random ...
Ali Taghavi's user avatar
4 votes
0 answers
145 views

Cyclic vectors for the translation operator

Let $b\in \mathbb{R}\neq 0$, and consider the translation operators: $$ \begin{align} T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\ f &\mapsto f(\cdot + b). \end{align} $$ *Are there known ...
ABIM's user avatar
  • 5,019
4 votes
0 answers
141 views

optimal regularity for elliptic pdes with $div(L^\infty)$ right-hand side (Hodge decomposition?)

Question: In a smooth, bounded domain $\Omega\subset \mathbb R^d$, is it true that solutions $\phi_f$ of $$ \begin{cases} -\Delta \phi_f=\operatorname{div}f & \mbox{in }\Omega\\ \phi_f = 0 & \...
leo monsaingeon's user avatar
7 votes
1 answer
173 views

Are $\log(\sigma(A(z))$ subharmonic functions?

Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$. Is it ...
Pritam Bemis's user avatar
3 votes
1 answer
264 views

Harmonic interpolation with analytic initial condition

Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold. Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function. Is there a Harmonic ...
Ali Taghavi's user avatar
2 votes
0 answers
148 views

About the probability of satisfying in a commutator type equation

Let $N$ be a closed normal subgroup of a compact group $G$. We denote the unique Haar measure of a compact group $A$ by $\mathbf m_A$, and drop $A$ if there exists no ambiguity. Fix $y\in G$. For $g\...
Meisam Soleimani Malekan's user avatar
6 votes
0 answers
267 views

Homogenized Mehta integral

For $a=(a_1,a_2)$ and $b=(b_1,b_2)$ in $\mathbb{C}^2$ let me use the notation from classical invariant theory $$ (ab)=a_1b_2-a_2b_1\ . $$ If I run out of letters $a,b,c,\ldots$, then I will switch to $...
Abdelmalek Abdesselam's user avatar
5 votes
0 answers
117 views

Good (Sidon) Approximation of "Bumps"

Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
user3293260's user avatar
2 votes
0 answers
178 views

Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$

Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$. For a fixed Littlewood-Paley decomposition $\chi \in \...
Desura's user avatar
  • 211
3 votes
0 answers
71 views

Density of the Mellin transform inside the direct integral of induced representations

I'm trying to better understand the continuous spectrum of $G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, which is the direct integral of induced representations $\mathbf H(s) = \operatorname{Ind}_{...
D_S's user avatar
  • 6,100
15 votes
3 answers
2k views

Integration of a function over 7-sphere

Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$. The problem is finding or approximating the ...
Hrushikesh Pawar's user avatar

1
5 6
7
8 9
29